Question

Find the mode of the following data.

Class interval	0-20	20-40	40-60	60-80	80-100	100-120	120-140
Frequency	6	8	10	12	6	5	3

Answer 1

Hint: From the question it can be interpreted that we need to find the mode of the grouped data. In general mode is the number that appears most frequently in a set of data or else it can also be said that for a grouped data the observation with maximum frequency is called the mode. The equation for mode of grouped data is given by
     $$Mode=l+\left({\displaystyle \frac{f_1-f_0}{2f_1-f_0-f_2}}\right)\times h$$
Where, l = lower limit of modal class
h= the size of class interval (assuming that the classes are of equal size.
$$f_1$$= the frequency of modal class
$$f_0$$= the frequency of the class preceding the modal class
$$f_2$$= the frequency of the class succeeding the modal class

Complete step-by-step answer:
Let us reconstruct according to our convenience

Class interval	Frequency
0-20	6
20-40	8
40-60	10
60-80	12
80-100	6
100-120	5
120-140	3

 Firstly, let us identify the modal class
For a grouped data with classes of equal size the class for which the frequency is the highest is known as modal class
In the grouped given to us maximum frequency is 12 of the class 60-80
So, modal class is 60-80
Lower limit of modal class= l = 60
The size of class interval = h =80-60=20
The frequency of modal class= $$f_1$$= 12
The class preceding the modal class is 40-60
From the table, the frequency of the class preceding the modal class=$$f_0$$= 10
The class succeeding the modal class 80-100
From the table, the frequency of the class succeeding the modal class =$$f_2$$ =6
Now let us substitute the values in the given formula
$$Mode=l+\left({\displaystyle \frac{f_1-f_0}{2f_1-f_0-f_2}}\right)\times h$$
$$\begin{array}{rl}& \Rightarrow Mode=60+\left({\displaystyle \frac{12-10}{2\left(12\right)-10-6}}\right)\times 20\\ & \Rightarrow Mode=60+\left({\displaystyle \frac{2}{8}}\right)\times 20=60+5\\ & \Rightarrow Mode=65\end{array}$$
Therefore the mode of the given grouped data is 65

Note: Students should observe and identify whether the data given is grouped or ungrouped. If the data is ungrouped then the observation which is repeated the maximum number of times will be the mode of the given data. The values of$$f_1$$,$$f_0$$,$$f_2$$ should be noted down precisely without any errors.